The fastest recorded pitch in Major League Baseball was thrown by Nolan Ryan in 1974. If this pitch were thrown horizontally, the ball would fall 0.809 m (2.65 ft) by the time it reached home plate, 18.3 m (60 ft) away.
The acceleration of gravity is 9.81 m/s2 . How fast was Ryan’s pitch? Answer in units of m/s
45.03756158
To determine the speed of Nolan Ryan's pitch, we can use the equation for vertical motion:
Δy = vi * t + 0.5 * a * t^2
Where:
- Δy is the vertical distance traveled (0.809 m)
- vi is the initial vertical velocity (unknown)
- t is the time of flight (unknown)
- a is the acceleration due to gravity (-9.81 m/s^2)
Since we know that the pitch traveled horizontally for a distance of 18.3 m and there is no vertical displacement in the horizontal direction, we can use the equation for horizontal motion:
Δx = vi * t
Where:
- Δx is the horizontal distance traveled (18.3 m)
- vi is the initial horizontal velocity (unknown)
- t is the same time of flight as before
By rearranging this equation, we can solve for t:
t = Δx / vi
Substituting this value for t in the vertical motion equation:
0.809 m = (Δx / vi) * t + 0.5 * a * t^2
Simplifying the equation and rearranging for vi:
vi = Δx / sqrt((2 * Δy) / a)
Plugging in the given values:
vi = 18.3 m / sqrt((2 * 0.809 m) / 9.81 m/s^2)
Calculating the expression within the square root:
vi = 18.3 m / sqrt(1.618 m / 9.81 m/s^2)
vi = 18.3 m / sqrt(0.165)
vi = 18.3 m / 0.406
vi = 45.07 m/s
Therefore, Nolan Ryan's pitch had a speed of approximately 45.07 m/s.
To calculate the speed of Nolan Ryan's pitch, we need to find the initial velocity of the pitch. We can use the equation of motion:
d = vit + (1/2)at²
where:
- d is the distance traveled by the ball (60 ft = 18.3 m)
- vi is the initial velocity (unknown)
- a is the acceleration due to gravity (9.81 m/s²)
- t is the time it takes for the ball to reach home plate
The time it takes for the ball to reach home plate can be calculated using the equation:
d = vit + (1/2)at²
Rearranging the equation:
t = √((2d) / a)
Now, substitute the given values into the equation:
t = √((2 * 18.3) / 9.81)
t ≈ 1.42 seconds
Now that we have the time, we can find the initial velocity using the equation:
vi = (d - (1/2)at²) / t
Substitute the values:
vi = (18.3 - (1/2) * 9.81 * 1.42²) / 1.42
vi ≈ 32.37 m/s
Therefore, Nolan Ryan's pitch was approximately 32.37 m/s.
distance fell= 1/2 g t^2
.809=4.9 t^2 solve for t, the time it was in the air.
howfast=distance/time= 18.3m/time